Categories: Quantum information.

Quantum gate

In quantum computing, quantum gates are the equivalent of classical binary logic gates such as \(\mathrm{NOT}\), \(\mathrm{AND}\), etc. Because of the continuous nature of qubits, the number of possible quantum gates is uncountably infinite, so we only consider the most important examples here.

One-qubit gates

As an example, consider the following must general single-qubit state \(\ket{\psi}\):

\[\begin{aligned} \ket{\psi} = \alpha \ket{0} + \beta \ket{1} = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \end{aligned}\]

Arguably the most famous and/or most fundamental quantum gates are the Pauli matrices:

\[\begin{aligned} \boxed{ X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} } \qquad \boxed{ Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} } \qquad \boxed{ Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} } \end{aligned}\]

They have the following effect on \(\ket{\psi}\). Note that \(X\) is equivalent to the classical \(\mathrm{NOT}\) gate (and is often given that name), and \(Z\) is sometimes called the phase-flip gate:

\[\begin{aligned} X \ket{\psi} = \begin{bmatrix} \beta \\ \alpha \end{bmatrix} \qquad Y \ket{\psi} = \begin{bmatrix} -i \beta \\ i \alpha \end{bmatrix} \qquad Z \ket{\psi} = \begin{bmatrix} \alpha \\ -\beta \end{bmatrix} \end{aligned}\]

In fact, \(Z\) is a specific case of the phase shift gate \(R_\phi\), which modifies the qubit’s phase without changing its amplitudes. For an angle \(\phi\), it is given by:

\[\begin{aligned} \boxed{ R_\phi = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \phi} \end{bmatrix} } \end{aligned}\]

For \(\phi = \pi\), we recover the Pauli-\(Z\) gate. In general, the action of \(R_\phi\) is as follows:

\[\begin{aligned} R_\phi \ket{\psi} = \begin{bmatrix} \alpha \\ e^{i \phi} \beta \end{bmatrix} \end{aligned}\]

Two common special cases of \(R_\phi\) are \(\phi = \pi/2\) and \(\phi = \pi/4\), respectively called \(S\) and \(T\):

\[\begin{aligned} \boxed{ S = R_{\pi/2} = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} } \qquad \quad \boxed{ T = R_{\pi/4} = \frac{1}{\sqrt{2}} \begin{bmatrix} \sqrt{2} & 0 \\ 0 & 1 + i \end{bmatrix} } \end{aligned}\]

Finally, we have the Hadamard gate \(H\), which is defined as follows:

\[\begin{aligned} \boxed{ H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} } \end{aligned}\]

Its action consists of rotating the qubit by \(\pi\) around the axis \((X + Z) / \sqrt{2}\) of the Bloch sphere:

\[\begin{aligned} H \ket{\psi} = \frac{1}{\sqrt{2}} \begin{bmatrix} \alpha + \beta \\ \alpha - \beta \end{bmatrix} \end{aligned}\]

Notably, it maps the eigenstates of \(X\) and \(Z\) to each other, and is its own inverse (i.e. unitary):

\[\begin{aligned} H \ket{0} = \ket{+} \qquad H \ket{1} = \ket{-} \qquad H \ket{+} = \ket{0} \qquad H \ket{-} = \ket{1} \end{aligned}\]

The Clifford gates are a set including \(X\), \(Y\), \(Z\), \(H\) and \(S\), or more generally any gates that rotate by multiples of \(\pi/2\) around the Bloch sphere. This set is not universal, meaning that if we start from \(\ket{0}\), we can only reach \(\ket{0}\), \(\ket{1}\), \(\ket{+}\), \(\ket{-}\), \(\ket{+i}\) \(\ket{-i}\) using these gates.

If we add any non-Clifford gate, for example \(T\), then we can reach any point on the Bloch sphere, which means that the set is universal.

However, there is a problem: a qubit has an uncountable infinity of states, but a quantum circuit consists of a countably infinite sequence of gates, at most. Therefore, technically, we can never reach the whole Bloch sphere, but we can come up with circuits that approximate a target state to some degree \(\varepsilon\). This is the definition of universality: any state can be approximated.

Two-qubit gates

As an example, let us consider the following two pure one-qubit states \(\ket{\psi_1}\) and \(\ket{\psi_2}\):

\[\begin{aligned} \ket{\psi_1} = \alpha_1 \ket{0} + \beta_1 \ket{1} = \begin{bmatrix} \alpha_1 \\ \beta_1 \end{bmatrix} \qquad \quad \ket{\psi_2} = \alpha_2 \ket{0} + \beta_2 \ket{1} = \begin{bmatrix} \alpha_2 \\ \beta_2 \end{bmatrix} \end{aligned}\]

The composite state of both qubits, assuming they are pure, is then their tensor product \(\otimes\):

\[\begin{aligned} \ket{\psi_1 \psi_2} = \ket{\psi_1} \otimes \ket{\psi_2} &= \alpha_1 \alpha_2 \ket{00} + \alpha_1 \beta_2 \ket{01} + \beta_1 \alpha_2 \ket{10} + \beta_1 \beta_2 \ket{11} \\ &= c_{00} \ket{00} + c_{01} \ket{01} + c_{10} \ket{10} + c_{11} \ket{11} \end{aligned}\]

Note that a two-qubit system may be entangled, in which case the coefficients \(c_{00}\) etc. cannot be written as products, i.e. \(\ket{\psi_2}\) cannot be expressed separately from \(\ket{\psi_1}\), and vice versa.

In other words, the general action of a two-qubit quantum gate can be expressed in the basis of \(\ket{00}\), \(\ket{01}\), \(\ket{10}\) and \(\ket{11}\), but not always in the basis of \(\ket{0}_1\), \(\ket{1}_1\), \(\ket{0}_2\) and \(\ket{1}_2\).

With that said, the first two-qubit gate is \(\mathrm{SWAP}\), which simply swaps \(\ket{\psi_1}\) and \(\ket{\psi_2}\):

\[\begin{aligned} \boxed{ \mathrm{SWAP} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} } \end{aligned}\]

This matrix is given in the basis of \(\ket{00}\), \(\ket{01}\), \(\ket{10}\) and \(\ket{11}\). Note that \(\mathrm{SWAP}\) cannot generate entanglement, so if its input is separable, its output is too. In any case, its effect is clear:

\[\begin{aligned} \mathrm{SWAP} \ket{\psi_1 \psi_2} &= c_{00} \ket{00} + c_{10} \ket{01} + c_{01} \ket{10} + c_{11} \ket{11} \end{aligned}\]

Next, there is the controlled NOT gate \(\mathrm{CNOT}\), which “flips” (applies \(X\) to) \(\ket{\psi_2}\) if \(\ket{\psi_1}\) is true:

\[\begin{aligned} \boxed{ \mathrm{CNOT} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} } \end{aligned}\]

That is, it swaps the last two coefficients \(c_{10}\) and \(c_{11}\) in the composite state vector:

\[\begin{aligned} \mathrm{CNOT} \ket{\psi_1 \psi_2} &= c_{00} \ket{00} + c_{01} \ket{01} + c_{11} \ket{10} + c_{10} \ket{11} \end{aligned}\]

More generally, from every one-qubit gate \(U\), we can define a two-qubit controlled U gate \(\mathrm{CU}\), which applies \(U\) to \(\ket{\psi_2}\) if \(\ket{\psi_1}\) is true:

\[\begin{aligned} \boxed{ \mathrm{CU} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & u_{00} & u_{01} \\ 0 & 0 & u_{10} & u_{11} \end{bmatrix} } \end{aligned}\]

Where the lower-right 2x2 block is simply \(U\). The general action of this gate is given by:

\[\begin{aligned} \mathrm{CU} \ket{\psi_1 \psi_2} &= c_{00} \ket{00} + c_{01} \ket{01} + (c_{10} u_{00} + c_{11} u_{01}) \ket{10} + (c_{10} u_{10} + c_{11} u_{11}) \ket{11} \end{aligned}\]

A set of gates is universal if all possible mappings from \(n\) to \(n\) qubits can be approximated using only these gates. A minimal universal set is \(\{\mathrm{CNOT}, T, S\}\), and there exist many others.

References

  1. J.S. Neergaard-Nielsen, Quantum information: lectures notes, 2021, unpublished.
  2. S. Aaronson, Introduction to quantum information science: lecture notes, 2018, unpublished.

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